Two concentric coils, each of radius equal to 2π cm, are placed at right angles to each other. Currents of 3 A and 4 A respectively are flowing through the two coils. The magnetic induction at the center of the coils will be

50 μT

The correct answer is Option (1) → 50 μT

Given:

Radius of each coil, R = 2π cm = 0.02π m

Currents, I₁ = 3 A, I₂ = 4 A

Magnetic field at the center of a circular coil:

$B = \frac{\mu_0 I}{2R}$

Since the coils are at right angles, the magnetic fields are perpendicular. Total magnetic induction:

$B_{\text{total}} = \sqrt{B_1^2 + B_2^2}$

Calculate B₁ and B₂:

$B_1 = \frac{\mu_0 I_1}{2 R} = \frac{4\pi \times 10^{-7} \cdot 3}{2 \cdot 0.02\pi} = \frac{12 \pi \times 10^{-7}}{0.04 \pi} = 3 \times 10^{-5}\ \text{T}$

$B_2 = \frac{\mu_0 I_2}{2 R} = \frac{4\pi \times 10^{-7} \cdot 4}{2 \cdot 0.02\pi} = \frac{16 \pi \times 10^{-7}}{0.04 \pi} = 4 \times 10^{-5}\ \text{T}$

Total magnetic field:

$B_{\text{total}} = \sqrt{(3 \times 10^{-5})^2 + (4 \times 10^{-5})^2} = \sqrt{9 + 16} \times 10^{-10} = 5 \times 10^{-5}\ \text{T}$

∴ Magnetic induction at the center = 5 × 10⁻⁵ T